How rigorously this is implemented varies and I don't know much about US policy in this respect but there are actually well-established international rules on this, in particular through the Geneva Convention. The US is only party to the 1967 New York protocol but it includes very similar rules.
Both are based on a similar logic, listing a number of basic rights but, instead of defining specific rules, it anchors the treatment of refugees to that of other groups of people (depending on the article either nationals or other aliens in the country). For example, here is the article on religion:
The Contracting States shall accord to refugees within their territories treatment at least as favourable as that accorded to their nationals with respect to freedom to practice their religion and freedom as regards the religious education of their children.
And here is the beginning of the article on “wage-earning employment”:
The Contracting State shall accord to refugees lawfully staying in their
territory the most favourable treatment accorded to nationals of a foreign
country in the same circumstances, as regards the right to engage in wage-earning employment.